AP Calc Quiz

The graph of a function f is shown above. If g is the function defined by g(x) = (x^2 +1)/f(x), what is the value of g'(2)

- 8/9

∫(x^2+1)/(x^3+3x-5)

-1/6 × 1/(x^3+3x-5)^2 + C

The figure above shows the graphs of the functions f and g. If h(x) = f(x)g(x), then h'(2) =

-11/2

Let 𝑓 be a continuous function such that ⅆ∫017𝑓(𝑥)ⅆ𝑥=8, ⅆ∫1720𝑓(𝑥)ⅆ𝑥=−3, and ⅆ∫1320𝑓(𝑥)ⅆ𝑥=7. What is the value of ⅆ∫013𝑓(𝑥)ⅆ𝑥?

-2

∫1→-1 (x^2-x)/x dx is

-2

The function 𝑓 is an antiderivative of the function 𝑔 defined by 𝑔(𝑥)=3−√𝑥^2+𝑥+4cos⁡𝑥. Which of the following is the 𝑥-coordinate of the location of a local maximum for the graph of 𝑦=𝑓(𝑥) ?

-2.161

If 𝑓(𝑥)=sin(𝑥^2+𝜋), then f'(√2π)=

-2√2π

If f is the function given by f(x) = 3x^2 − x^3, then the average rate of change of f on the closed interval [1, 5 ] is

-3

Let H(x) be an antiderivative of (𝑥^3+sin𝑥)/(𝑥^2+2). If H(5) = 𝜋, then H(2) =

-5.867

The number of gallons of water in a storage tank at time t, in minutes, is modeled by w(t) = 25 − t2 for 0≤𝑡≤5. At what rate, in gallons per minute, is the amount of water in the tank changing at time t = 3 minutes?

-6

The graph of the piecewise linear function 𝑓 is shown above. What is the value of ⅆ∫012𝑓′(𝑥)ⅆ𝑥 ?

-8

lim(x→∞) (10-6x^2)/(5+3e^x) is

0

Let f be the function defined by f(x) = ln(x^2 + 1), and let g be the function defined by g(x) = x^5 + x^3. The line tangent to the graph of f at x = 2 is parallel to the line tangent to the graph of g at x = a, where a is a positive constant. What is the value of a ?

0.447

Let f be a function with derivative given by 𝑓′(𝑥)=𝑥^3−8𝑥^2+3/(𝑥^3+1) for -1<𝑥<9. At what value of x does f attain a relative maximum? Responses

0.638

limx→0 sinx/(e^x-1) is

1

At time 𝑡, 0<𝑡<2, the velocity of a particle moving along the 𝑥-axis is given by 𝑣(𝑡)=𝑡sin⁡(𝑡3). Let 𝑡=𝑏 be the time at which the particle changes direction from moving left to moving right. What is the total distance traveled by the particle during the time interval 0<𝑡<𝑏 ?

1.011

The base of a solid is the region in the first quadrant bounded by the 𝑦-axis, the 𝑥-axis, the graph of 𝑦=𝑒^𝑥, and the vertical line 𝑥=1. For this solid, each cross section perpendicular to the 𝑥-axis is a square. What is the volume of the solid? Responses

1/2e^2-1/2

If 𝑓(𝑥)=ln𝑥, then lim𝑥→3𝑓(𝑥)−𝑓(3)/(𝑥−3) is

1/3

Tara's heart rate during a workout is modeled by the differentiable function ℎ, where ℎ(𝑡) is measured in beats per minute and 𝑡 is measured in minutes from the start of the workout. Which of the following expressions gives Tara's average heart rate from 𝑡=30 to 𝑡=60 ?

1/30 ∫60→30h(t)dt

∫(1/3x+12)dx =

1/3ln|x+4|+C

What is the value of 𝑥 at which the minimum value of 𝑦=3𝑥^4/3−2𝑥 occurs on the closed interval [0,1] ?

1/8

The number of bacteria in a container increases at the rate of R(t) bacteria per hour. If there are 1000 bacteria at time t = 0, which of the following expressions gives the number of bacteria in the container at time t = 3 hours?

1000+∫3→0 R(t)dt

The weight of a population of yeast is given by a differentiable function y, where y(t) is measured in grams and t is measured in days. The weight of the yeast population increases according to the equation 𝑑𝑦/𝑑𝑡=𝑘𝑦, where k is a constant. At time t = 0, the weight of the yeast population is 120 grams and is increasing at the rate of 24 grams per day. Which of the following is an expression for y(t) ?

120e^t/5

The table above gives the velocity v(t), in miles per hour, of a truck at selected times t, in hours. Using a trapezoidal sum with the three subintervals indicated by the table, what is the approximate distance, in miles, the truck traveled from time t = 0 to t = 3 ?

130

If dy/dx = 2-y, and if y = 1 when x = 1, the y =

2 = e^(1-x)

Let 𝑓 be the function defined above, where 𝑘 is a positive constant. For what value of 𝑘, if any, is 𝑓 continuous? Responses

2.081

A particle travels along a straight line with velocity 𝑣(𝑡)=3𝑒^−𝑡/2 * sin(2𝑡) meters per second. What is the total distance, in meters, traveled by the particle during the time interval 0≤𝑡≤2 seconds? Responses

2.261

Let 𝑔 be a twice-differentiable function with 𝑔′(𝑥)>0 and 𝑔″(𝑥)>0 for all real numbers 𝑥, such that 𝑔(3)=12 and 𝑔(5)=18. Which of 20, 21, and 22 are possible values for 𝑔(6) ?

22 only

If f(x) = (2x2 +5)7, then f'(x) =

28x(2x2 + 5)6

For any real number x, limh→0 (sin(2(x+h))-sin(2x)/h =

2cos(2x)

A particle moves along a straight line so that at time t≥0 its acceleration is given by a(t) = 12t. At time t = 0, the velocity of the particle is 2 and the position of the particle is 5. Which of the following is an expression of the position of the particle at time t≥0?

2t^3 + 2t + 5

d/dx (x^3sec(2x))

2x^3sec(2x)tan(2x)+3x^2sec(2x)

An isosceles right triangle with legs of length s has area 𝐴=12𝑠^2. At the instant when 𝑠=√32 centimeters, the area of the triangle is increasing at a rate of 12 square centimeters per second. At what rate is the length of the hypotenuse of the triangle increasing, in centimeters per second, at that instant?

3

limx→∞ ln(e^3x + x)/3

3

The function 𝑔 is differentiable and satisfies 𝑔(−1)=4 and 𝑔′(−1)=2. What is the approximation of 𝑔(−1.2) using the line tangent to the graph of 𝑔 at 𝑥=−1 ?

3.6

A person stands 30 feet from point 𝑃 and watches a balloon rise vertically from the point, as shown in the figure above. The balloon is rising at a constant rate of 2 feet per second. What is the rate of change, in radians per second, of angle 𝜃 at the instant when the balloon is 40 feet above point P?

3/125

The region enclosed by the graphs of y = x^2 and y = 2x is the base of a solid. For the solid, each cross section perpendicular to the y-axis is a rectangle whose height is 3 times its base in the xy-plane. Which of the following expressions gives the volume of the solid?

3∫4→0(√y = y/2)^2 dy

Let 𝑔 be the function with first derivative 𝑔′(𝑥)=√𝑥^3+𝑥 for 𝑥>0. If 𝑔(2)=−7, what is the value of 𝑔(5) ?

4.402

If x + 3y^1/3 = y, what is dy/dx at the point (2,8)

4/3

Let R be the region bounded by the graphs of y = 2x and y = 4x - x^2. What is the area of R?

4/3

If 𝑥^2+𝑥𝑦−3𝑦=3, then at the point (2, 1), dy/dc

5

Let g be the function given be g(s) = ∫x→3(t^2 - 5t -14)dt. What is the x-coordinate of the point of inflection of the graph of g?

5/2

∫2→0 (x^3+1)^1/2 x^2 dx =

52/9

d/dx (x^5 - 5^x)

5x^4-(ln5)5^x

For a certain continuous function f, the right Riemann sum approximation of ∫2→0𝑓(𝑥) 𝑑𝑥 with n subintervals of equal length is 2(𝑛+1)(3𝑛+2)/𝑛^2 for all n. What is the value of ∫2→0𝑓(𝑥) 𝑑𝑥?

6

The height of an object at time 𝑡≥1 is given by ℎ(𝑡)=𝑡2−16𝑡+15. What is the velocity of the object at time 𝑡=3 ?

7.778

If ∫4→−10𝑔(𝑥)𝑑𝑥=−3 and ∫4→6𝑔(𝑥)𝑑𝑥=5, then ∫6→-10𝑔(𝑥)𝑑𝑥=

8

If the average value of a continuous function f on the interval [−2, 4 ] is 12, what is ∫4→-2 𝑓(𝑥)/8 𝑑𝑥?

9

A file is downloaded to a computer at a rate modeled by the differentiable function f(t), where t is the time in seconds since the start of the download and f(t) is measured in megabits per second. Which of the following is the best interpretation of f'(5) = 2.8 ?

At time t = 5 seconds, the rate at which the file is downloaded to the computer is increasing at a rate of 2.8 megabits per second per second.

The graph of f', the derivative of the function f, is shown in the figure above. Which of the following statements must be true? I. f is continuous on the open interval (a, b). II. f is decreasing on the open interval (a, b). III. The graph of f is concave down on the open interval (a, b).

I and III only

For any function f, which of the following statements must be true? I. If f is defined at x = a, then lim𝑥→𝑎𝑓(𝑥)=𝑓(𝑎). II. If f is continuous at x = a, then lim𝑥→𝑎𝑓(𝑥)=𝑓(𝑎). III. If f is differentiable at x = a, then lim𝑥→𝑎𝑓(𝑥)=𝑓(𝑎). Responses

II and III only

How many vertical asymptotes does the graph of 𝑦=𝑥−2𝑥4−16 have?

One

Let f be a continuous function for all real numbers. Let g be the function defined by 𝑔(𝑥)=∫𝑥→1𝑓(𝑡)𝑑𝑡. If the average rate of change of g on the interval 2≤𝑥≤5 is 6, which of the following statements must be true? Responses

The average value of f on the interval 2≤𝑥≤5 is 6.

A particle moves along the y-axis so that at time t≥0 its position is given by y(t) = t^3 - 4t^2 + 4t +3. Which of the following statement describes the motion of the particle at time t = 1?

The particle is moving down the 𝑦-axis with decreasing velocity.

The table above gives values of a continuous function 𝑓 at selected values of 𝑥. Based on the information in the table, which of the following statements must be true?

There exists a value 𝑐, where −5<𝑐<2, such that 𝑓(𝑐)=4.

Let 𝑓 be the function defined by 𝑓(𝑥)=1/4𝑥^4−2/3𝑥^3+1/2𝑥^2−1/2𝑥. For how many values of 𝑥 in the open interval (0,1.565) is the instantaneous rate of change of 𝑓 equal to the average rate of change of 𝑓 on the closed interval [0,1.565] ?

Three

At time 𝑡=0, a storage tank is empty and begins filling with water. For 𝑡>0 hours, the depth of the water in the tank is increasing at a rate of 𝑊(𝑡) feet per hour. Which of the following is the best interpretation of the statement 𝑊′(2)>3 ?

Two hours after the tank begins filling with water, the rate at which the depth of the water is rising is increasing at a rate greater than 3 feet per hour per hour.

The function 𝑓 is continuous on the closed interval [0,5]. The graph of 𝑓′, the derivative of 𝑓, is shown above. On which of the following intervals is 𝑓 increasing?

[0,2] and [4.5]

For what value of 𝑏 does the integral ⅆ∫1𝑏𝑥2ⅆ𝑥 equal lim𝑛→∞∑𝑘=1𝑛(1+2𝑘𝑛)^22𝑛 ?

b = 3 only

Shown above is a slope field for which of the following differential equations?

dy/dx = y^2(4-y)/4

Shown above is a slope field for which of the following differential equationss?

dy/dx = y^3

The twice-differentiable functions f, g, and h have second derivatives given above. Which of the functions f, g, and h have a graph with exactly two points of inflection?

f and g only

Let f be the function given by 𝑓(𝑥)=𝑥−22|𝑥−2|. Which of the following is true?

f has a jump discontinuity at x = 2 .

Let f be the function defined above. Which of the following statements about f is true?

f is continuous and differentiable at x = 5.

The table above gives values of a function f at selected values of x. If f is twice-differentiable on the interval 1≤𝑥≤5, which of the following statements could be true?

f' is negative and increasing for 1≤𝑥≤5.

The graph of a differentiable function f is shown in the figure above. Which of the following is true?

f'(3) < f'(0) < f'(-2)

The function g is continuous on the closed interval [1, 4] with g(1) = 5 and g(4) = 8. Of the following conditions, which would guarantee that there is a number c in the open interval (1, 4) where g'(c) = 1 ?

g is differentiable on the open interval (1, 4).

The continuous function f is positive and has domain x > 0. If the asymptotes of the graph of f are x = 0 and y = 2, which of the following statements must be true?

lim→0+ f(x)=∞ and limf(x)→∞ f(x) = 2

The graphs of the functions f and g are shown in the figures above. Which of the following statements is false?

lim𝑥→1(𝑓(𝑥)𝑔(𝑥+1)) does not exist.

Let F be the function given by 𝐹(𝑥)=∫𝑥→3(tan(5𝑡)sec(5𝑡)−1) 𝑑𝑡. Which of the following is an expression for F'(x) ?

tan(5x)sec(5x)−1

∫x^4/4 dx

x^3/12 + C

Which one of the following is an equation of the line tangent to the graph of y = cosx at x = π/2?

y = -x + π/2

The graph of which of the following functions has exactly one horizontal asymptote and no vertical asymptotes?

y = 1/(x^2 +1)

The table above gives values of the differentiable function 𝑓 and its derivative at selected values of 𝑥. If 𝑔 is the inverse function of 𝑓, which of the following is an equation of the line tangent to the graph of 𝑔 at the point where 𝑥=2 ?

y = 1/5(x-2) + 3

Which of the following is the solution to the differential equation 𝑑𝑦/𝑑𝑥=2𝑦/(2𝑥+1) with the initial condition 𝑦(0)=𝑒 for 𝑥>−1/2?

y = 2ex + e

If f is the function given by f(x) = ex/3, which of the following is an equation of the line tangent to the graph of f at the point (3 ln 4, 4) ?

y = 4 = 4/3(x-3ln4)

The equation 𝑦=2𝑒^(6𝑥−5) is a particular solution to which of the following differential equations? Responses

y' - 6y - 30 = 0

Let f be the function given by f(x) = 2 cos x + 1. What is the approximation for f(1.5) found by using the line tangent to the graph of f at x = π/2 ?

π - 2

A particle moves along the x-axis so that at time t > 0 its position is given by x(t) = 12e^−t sin t. What is the first time t at which the velocity of the particle is zero?

π/4

The graph of a function f is shown above. Which of the following expresses the relationship between ∫0→2𝑓(𝑥)𝑑𝑥, ∫0→3𝑓(𝑥)𝑑𝑥, and ∫2→3𝑓(𝑥)𝑑𝑥?

∫3→2𝑓(𝑥)𝑑𝑥 <∫3→0𝑓(𝑥)𝑑𝑥 < ∫2→0𝑓(𝑥)𝑑𝑥?

If ∫4→1f(x)dx = 8 and ∫4→1g(x)dx = -2, which of the following cannot be determined from the information given? Responses

∫4→1(3f(x)dx)

Let f be the function defined by f(x) = -3 + 6x2 − 2x3. What is the largest open interval on which the graph of f is both concave up and increasing?

(0,1)

The function f is given by f(x) = 4x^3 - x^4. On what intervals is the graph of f concave up?

(0,2)

The function f has first derivative given by f'(x) = x^4 − 6x^2 − 8x − 3. On what intervals is the graph of f concave up?

(2, ∞) only

d/dx (2(sin√x)^2)

(2sin√xcos√x)/√x

If 𝑓(𝑥)=)5−𝑥)/(𝑥^3+2), then f'(x) =

(2x^3-15x^2-2)/(x^3+2)^2

The table above gives selected values for the differentiable function f. In which of the following intervals must there be a number c such that f'(c) = 2?

(8,12)